Introduction:

             Multiplication is the shortest method of adding same numbers for more than once. For example, by adding the numbers 3 + 3 + 3 + 3 , we get 12. Which can be stated that, by adding 3 for four  times we get 12. This can also be written as 3 x 4 = 12. We can use the same procedure for the addition of fractions. In this section let us discuss about the multiplication of fraction with whole numbers.



Multiplication of proper fraction by a whole number:


Let us look at the following diagram, where the first three diagrams show the fractions $\frac{1}{4}$ and the last diagram shows the fraction $\frac{3}{4}$.

fraction11

By adding, the functions $\frac{1}{4}$ , three times , we have,

       $\frac{1}{4}+\frac{1}{4}+\frac{1}{4}$ = $\frac{1}{4}\times 3 $

                                                         =$\frac{1\times3}{4}$

                                                        = $\frac{3}{4}$

Hence we have the following steps while multiplying a fraction by a whole number.
Step 1: Rewrite the whole number in a fraction form with 1 as the denominator.
Step 2: Multiply the numbers in the numerator.
Step 3: Multiply the numbers in the denominator.
Step 4: If the fraction obtained from steps 2 and 3 has common factor, Divide the numerator and the denominator by the Highest Common Factor of the
            numerator and the denominator.
Step 5: If the simplest form of the fraction obtained from step 4 is a proper fraction, highlight this as the final answer else
            convert it into mixed fraction and highlight the answer.

Example 1: Multiply $\frac{10}{3}\times 12$

Solution: We have, $\frac{10}{3}\times 12$

                                              = $\frac{10}{3}\times \frac{12}{1}$ [ writing the whole number in fraction form 12/1 ]

                                              = $\frac{10\times 12}{3\times 1}$ [ combining the numerators and the denominators ]

                                              =$\frac{120}{3}$ [ writing the product of the numbers in the numerator and the denominator ]

                                              = 40 Final answer obtained by dividing 120 by 3 .

Example 2: $\frac{5}{24}\times 4$


Solution: We have, $\frac{5}{24}\times 4$

                                             =$\frac{5}{24}\times \frac{4}{1}$ [ expressing the whole number in fractional form 4/1 ]

                                             =$\frac{5\times 4}{24\times 1}$ [combining the numerators and the denominators ]

                                             =$\frac{20}{24}$ [ writing the product of the numbers in the numerator and the denominator ]

                                             =$\frac{20\div 4}{24\div 4}$ [ dividing the numerator and the denominator by the HCF 4 ]

                                             =$\frac{5}{6}$ Final answer in the simplest form.
Working:
To find the HCF (Highest Common Factor ) of 20 and 24
Let us first express the two numbers as product of primes.
fraction12
20 = 2 x 2 x 5
24 = 2 x 2 x 2 x 3
Highest Common Factor ( HCF ) = 2 x 2 = 4

Word problem: Sam reads a book for $5\frac{1}{2}$ hours a day. How many hours will be be reading if he completes the book in 6 days.

Solution: In the given problem , Number of hours read in 1 day = $5\frac{1}{2}$
Therefore the number of hours read in 6 days = $5\frac{1}{2} \times 6$

                                                                 =$\frac{11}{2}\times\frac{6}{1}$

                                                                 =$\frac{11\times6}{2\times1}$

                                                                =$\frac{66}{2}$

                                                                = 33 hrs

                                  (i.e) Sam will be completing the book in 33 hrs                

Practice Questions:

Multiply the following and express your answer in simplest form.

1. $\frac{2}{15}\times 20$

2. $\frac{12}{7}\times 6$

3. $1\frac{2}{3}\times 5$

*****